The problem in the image involves finding the average cost of production based on the marginal cost function. The marginal cost function provided is:

$\frac{dC}{dq} = 0.003q^2 - 0.4q + 40$

where $q$ is the number of units produced. We are also given additional information:

• The marginal cost is $27.50 when $q = 50$.
• Fixed costs are $5000.
• We need to find the average cost of producing 100 units.

Steps to solve:

1. Integrate the marginal cost function to find the total cost function $C(q)$.
2. Use the given condition $MC(50) = 27.50$ to determine the constant of integration.
3. Add the fixed cost to the cost function.
4. Calculate the total cost at $q = 100$.
5. Find the average cost by dividing the total cost by 100.

Let me go ahead and calculate that.The total cost function, before adding the constant of integration, is:

$C(q) = 0.001q^3 - 0.2q^2 + 40q + C_0$

where $C_0$ is the constant of integration. Next, we will use the condition that the marginal cost at $q = 50$ is $27.50 to determine $C_0$. Let me calculate that.The given condition that the marginal cost at $q = 50$ is $27.50 holds true, as the marginal cost calculation confirms this value.

Now, I will determine the constant $C_0$ by considering the fixed costs of $5000 and evaluating the total cost function.The average cost of producing 100 units is $80.00.

Would you like a detailed explanation of any step or further clarification?

Here are five related questions you might find interesting:

1. How do fixed costs influence the total cost function?
2. How can we interpret the marginal cost function in terms of production efficiency?
3. What role does the constant of integration play in calculating total costs?
4. How does average cost differ from marginal cost?
5. What happens to average cost if production continues to increase?

Tip: When analyzing cost functions, always differentiate between marginal, total, and average costs, as they provide insights into different aspects of production efficiency.